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. 2018 Oct;156(4):141.
doi: 10.3847/1538-3881/aad95c. Epub 2018 Sep 6.

On the Dynamics of the Inclination Instability

Ann-Marie Madigan  1 , Alexander Zderic  1 , Michael McCourt  2 , Jacob Fleisig  1
Affiliations

On the Dynamics of the Inclination Instability

Ann-Marie Madigan et al. Astron J. 2018 Oct.

Abstract

Axisymmetric disks of eccentric Kepler orbits are vulnerable to an instability that causes orbits to exponentially grow in inclination, decrease in eccentricity, and cluster in their angle of pericenter. Geometrically, the disk expands to a cone shape that is asymmetric about the mid-plane. In this paper, we describe how secular gravitational torques between individual orbits drive this "inclination instability". We derive growth timescales for a simple two-orbit model using a Gauss N-ring code, and generalize our result to larger N systems with N-body simulations. We find that two-body relaxation slows the growth of the instability at low N and that angular phase coverage of orbits in the disk is important at higher N. As N → ∞, the e-folding timescale converges to that expected from secular theory.

Keywords: celestial mechanics-minor planets, asteroids; dynamical evolution and stability; general-planets and satellites.

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Figures

Figure 1.
Figure 1.
Idealized, “two-orbit” toy model for the inclination instability. The top right of each panel shows the location of the orbit in the disk from a face-on perspective. (top): Orbit 1 experiences a net upward force f. This force produces a torque along the b^ axis, rotating the orbital plane such that ia < 0. (bottom): Diametrically opposed orbit 2 feels a force due to the rotation of orbit 1; this force produces a torque along the a^ direction of orbit 2, rotating its orbital plane such that ib > 0.
Figure 2.
Figure 2.
(top): Median values of angles ia and ib for all particles in a disk as a function of time in units of orbital period. Shaded regions indicate one-sigma quantile contours. Both ia and ib rise exponentially at early times. As predicted by the two-orbit toy model, ia and ib increase with opposite sign. (bottom): Median value of orbital eccentricity for the same particles. As inclinations increase, eccentricities decrease.
Figure 3.
Figure 3.
Squared growth rate γ2 for two orbits gravitationally torquing each other. This is estimated from the linearized equations of motion for the system (equation 9), as a function of orbital eccentricity and angular separation between orbits in the plane, θ. Only the unstable region with γ2 > 0 is colored. The white, uncolored region of the plot with dashed contours is stable. The thick, gray band near the γ2 = 0 boundary marks the region in which the linear approximation in equation 5 breaks down, causing an order-unity uncertainty in our estimate of the timescale. This region also depends on the softening length we adopt.
Figure 4.
Figure 4.
(top): Exponential growth of the inclination angle ib for different numbers of disk particles (N) and fixed total mass Mdisk = 10−2M. Time is in units of orbital period at semimajor axis a = 1. Note that the slopes (growth rate) of the lines show a strong dependence on N. (bottom): Instability growth rate (γ) for different values of N and Mdisk. The expected secular growth rate (tsec−1) is plotted as dashed gray lines (corresponding from top to bottom to disk masses of Mdisk = [10−2, 10−3, 10−4]). Note that the growth rate’s dependence on N weakens as Mdisk is reduced, and ultimately converges to the secular growth rate.
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